Decision-Sufficient Posterior Approximation
Abstract
We investigate the consequences of requiring a posterior approximation to preserve a specified downstream decision problem. A target posterior P and loss determine a regret geometry on actions, a baseline approximation Q0 determines the forward-Kullback-Leibler information required to induce action changes, and a restricted approximation family 𝒬 determines which such changes are available. Contracting KL divergence over Bayes-action fibers gives exact distances to decision adequacy and decision failure together with the least-informative posterior deformations that reach either side of the decision boundary. In regular finite-dimensional problems, the target and baseline constructions have quadratic local limits: a target regret Hessian G and a baseline information metric JI . Their generalized eigenproblem Gv = γJIv orders local decision directions by regret consequence per unit information cost and induces a tolerance-dependent effective dimension. For restricted approximation families, the tangent image separates decision coverage from information efficiency: a family may miss consequential decision directions, or it may realize reachable directions only at excess Fisher cost. The resulting framework provides decision-relative criteria for comparing and designing posterior approximation families.
Related
- Bayesian Geometric Inversion for Wave Equation: Well-Posedness and Multiscale Multimodal Posterior Approximation
- Proximal Empirical Bayes for Sparse Regression with Posterior Decision Support
- A sufficient condition for the existence of solutions of singular linear-quadratic vector equations
- The Sufficient Set of Congruences of Binomial Coefficients for Primality
- Necessity of Ladyzhenskaya \& Ural'tseva's sufficient conditions in nonuniform ellipticity
BibTeX
@misc{plummer2026decisionsufficient,
title = {{Decision-Sufficient Posterior Approximation}},
author = {Sean Plummer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11038},
eprint = {2610.11038},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11038},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Decision-Sufficient Posterior Approximation
AU - Sean Plummer
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11038
UR - https://arxiv.org/abs/2610.11038
AB - We investigate the consequences of requiring a posterior approximation to preserve a specified downstream decision problem. A target posterior P and loss determine a regret geometry on actions, a baseline approximation Q_0 determines the forward-Kullback-Leibler information required to induce action changes, and a restricted approximation family 𝒬 determines which such changes are available. Contracting KL divergence over Bayes-action fibers gives exact distances to decision adequacy and decision failure together with the least-informative posterior deformations that reach either side of the decision boundary. In regular finite-dimensional problems, the target and baseline constructions have quadratic local limits: a target regret Hessian G and a baseline information metric J_I . Their generalized eigenproblem Gv = γJ_Iv orders local decision directions by regret consequence per unit information cost and induces a tolerance-dependent effective dimension. For restricted approximation families, the tangent image separates decision coverage from information efficiency: a family may miss consequential decision directions, or it may realize reachable directions only at excess Fisher cost. The resulting framework provides decision-relative criteria for comparing and designing posterior approximation families.
ER -